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Higher-order moment theories for dilute granular gases of smooth hard spheres

Published online by Cambridge University Press:  12 December 2017

Vinay Kumar Gupta Open the ORCID record for Vinay Kumar Gupta [Opens in a new window] ,
Priyanka Shukla Open the ORCID record for Priyanka Shukla [Opens in a new window]  and
Manuel Torrilhon
Vinay Kumar Gupta*
Affiliation:
SRM Research Institute and Department of Mathematics, SRM Institute of Science and Technology, Chennai 603203, India
Priyanka Shukla*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
Manuel Torrilhon
Affiliation:
Center for Computational Engineering Science, Department of Mathematics, RWTH Aachen University, Schinkelstr. 2, D-52062 Aachen, Germany
*
†Email addresses for correspondence: vinay.libra.gupta@gmail.com, priyanka@iitm.ac.in
†Email addresses for correspondence: vinay.libra.gupta@gmail.com, priyanka@iitm.ac.in
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Abstract

Grad’s method of moments is employed to develop higher-order Grad moment equations – up to the first 26 moments – for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff’s law while the other higher-order moments decay on a faster time scale. The nonlinear terms of the fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have a negligible effect on Haff’s law. Furthermore, an even larger Grad moment system, which includes the fully contracted sixth moment, is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of the scalar sixth-order moment into the Grad moment system has a negligible effect on Haff’s law. The constitutive relations for the stress and heat flux (i.e. the Navier–Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via the Chapman–Enskog expansion and via computer simulations. The linear stability of the homogeneous cooling state is analysed through the Grad 26-moment system and various subsystems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution, while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber, in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment theory is in excellent agreement with that obtained through existing theories.

JFM classification

Granular media: Granular media Rarefied Gas Flow: Kinetic theory Rarefied Gas Flow: Rarefied Gas Flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 451 - 501
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Copyright
© 2017 Cambridge University Press 

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References

Alam, M., Chikkadi, V. & Gupta, V. K. 2009 Density waves and the effect of wall roughness in granular Poiseuille flow: simulation and linear stability. Eur. Phys. J. ST 179, 69–90.CrossRefGoogle Scholar
Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon.Google Scholar
Bisi, M., Spiga, G. & Toscani, G. 2004 Grad’s equations and hydrodynamics for weakly inelastic granular flows. Phys. Fluids 16, 4235–4247.CrossRefGoogle Scholar
Bizon, C., Shattuck, M. D., Swift, J. B. & Swinney, H. L. 1999 Transport coefficients for granular media from molecular dynamics simulations. Phys. Rev. E 60, 4340–4351.Google Scholar
Bobylev, A. V. 1982 The Chapman–Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys. Dokl. 27, 29–31.Google Scholar
Breu, A. P. J., Ensner, H.-M., Kruelle, C. A. & Rehberg, I. 2003 Reversing the Brazil-nut effect: competition between percolation and condensation. Phys. Rev. Lett. 90, 014302.Google Scholar
Brey, J. J., Dufty, J. W., Kim, C. S. & Santos, A. 1998a Hydrodynamics for granular flow at low density. Phys. Rev. E 58, 4638–4653.Google Scholar
Brey, J. J., Dufty, J. W. & Santos, A. 1997 Dissipative dynamics for hard spheres. J. Stat. Phys. 87, 1051–1066.Google Scholar
Brey, J. J., Moreno, F. & Dufty, J. W. 1996 Model kinetic equation for low-density granular flow. Phys. Rev. E 54, 445–456.Google Scholar
Brey, J. J., Ruiz-Montero, M. J., Maynar, P. & García de Soria, M. I. 2005 Hydrodynamic modes, Green–Kubo relations, and velocity correlations in dilute granular gases. J. Phys.: Condens. Matter 17, S2489.Google Scholar
Brey, J. J., Ruiz-Montero, M. J. & Moreno, F. 1998b Instability and spatial correlations in a dilute granular gas. Phys. Fluids 10, 2976–2982.Google Scholar
Brilliantov, N. V. & Pöschel, T. 2004 Kinetic Theory of Granular Gases. Oxford University Press.Google Scholar
Campbell, C. S. 1990 Rapid granular flows. Annu. Rev. Fluid Mech. 22, 57–90.Google Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.Google Scholar
Corwin, E. I., Jaeger, H. M. & Nagel, S. R. 2005 Structural signature of jamming in granular media. Nature 435, 1075–1078.CrossRefGoogle ScholarPubMed
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Garzó, V. 2005 Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, 021106.Google Scholar
Garzó, V. 2013 Grad’s moment method for a granular fluid at moderate densities: Navier–Stokes transport coefficients. Phys. Fluids 25, 043301.Google Scholar
Garzó, V. & Dufty, J. W. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895–5911.Google Scholar
Garzó, V. & Santos, A. 2003 Kinetic Theory of Gases in Shear Flows. Nonlinear Transport. Kluwer Academic Publishers.CrossRefGoogle Scholar
Garzó, V., Santos, A. & Montanero, J. M. 2007 Modified Sonine approximation for the Navier–Stokes transport coefficients of a granular gas. Physica A 376, 94–107.Google Scholar
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–293.Google Scholar
Goldshtein, A. & Shapiro, M. 1995 Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations. J. Fluid Mech. 282, 75–114.Google Scholar
Grad, H. 1949a Note on N-dimensional Hermite polynomials. Commun. Pure Appl. Maths 2, 325–330.Google Scholar
Grad, H. 1949b On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths 2, 331–407.Google Scholar
Gupta, V. K.2011 Kinetic theory and Burnett order constitutive relations for a smooth granular gas. Master’s thesis, JNCASR, Bangalore, India.Google Scholar
Gupta, V. K.2015 Mathematical modeling of rarefied gas mixtures. PhD thesis, RWTH Aachen University, Germany.Google Scholar
Gupta, V. K. & Torrilhon, M. 2012 Automated Boltzmann collision integrals for moment equations. AIP Conf. Proc. 1501, 67–74.Google Scholar
Gupta, V. K. & Torrilhon, M. 2015a Comparison of relaxation phenomena in binary gas-mixtures of Maxwell molecules and hard spheres. Comput. Maths Applics. 70, 73–88; reprint: ibid, 72, 271–287 (2016).Google Scholar
Gupta, V. K. & Torrilhon, M. 2015b Higher order moment equations for rarefied gas mixtures. Proc. R. Soc. Lond. A 471, 20140754.Google Scholar
Haff, P. K. 1983 Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430.Google Scholar
Jenkins, J. T. & Richman, M. W. 1985a Grad’s 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355–377.Google Scholar
Jenkins, J. T. & Richman, M. W. 1985b Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys. Fluids 28, 3485–3494.CrossRefGoogle Scholar
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202.CrossRefGoogle Scholar
Khalil, N., Garzó, V. & Santos, A. 2014 Hydrodynamic Burnett equations for inelastic Maxwell models of granular gases. Phys. Rev. E 89, 052201.Google Scholar
Kremer, G. M. 2010 An Introduction to the Boltzmann Equation and Transport Processes in Gases. Springer.Google Scholar
Kremer, G. M. & Marques, W. Jr. 2011 Fourteen moment theory for granular gases. Kinet. Relat. Models 4, 317–331.Google Scholar
Kremer, G. M., Santos, A. & Garzó, V. 2014 Transport coefficients of a granular gas of inelastic rough hard spheres. Phys. Rev. E 90, 022205.Google Scholar
Kudrolli, A., Wolpert, M. & Gollub, J. P. 1997 Cluster formation due to collisions in granular material. Phys. Rev. Lett. 78, 1383–1386.Google Scholar
Liss, E. D., Conway, S. L. & Glasser, B. J. 2002 Density waves in gravity-driven granular flow through a channel. Phys. Fluids 14, 3309–3326.Google Scholar
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140, 223–256.Google Scholar
Lutsko, J. F. 2005 Transport properties of dense dissipative hard-sphere fluids for arbitrary energy loss models. Phys. Rev. E 72, 021306.Google Scholar
Maxwell, J. C. 1879 On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
McNamara, S. 1993 Hydrodynamic modes of a uniform granular medium. Phys. Fluids A 5, 3056–3070.Google Scholar
Melo, F., Umbanhowar, P. B. & Swinney, H. L. 1995 Hexagons, kinks, and disorder in oscillated granular layers. Phys. Rev. Lett. 75, 3838–3841.Google Scholar
Mitrano, P. P., Dahl, S. R., Cromer, D. J., Pacella, M. S. & Hrenya, C. M. 2011 Instabilities in the homogeneous cooling of a granular gas: a quantitative assessment of kinetic-theory predictions. Phys. Fluids 23, 093303.Google Scholar
Montanero, J. M., Santos, A. & Garzó, V. 2005 DSMC evaluation of the Navier–Stokes shear viscosity of a granular fluid. AIP Conf. Proc. 762, 797–802.Google Scholar
Mueth, D. M., Debregeas, G. F., Karczmar, G. S., Eng, P. J., Nagel, S. R. & Jaeger, H. M. 2000 Signatures of granular microstructure in dense shear flows. Nature 406, 385–389.Google Scholar
Mullin, T. 2000 Coarsening of self-organized clusters in binary mixtures of particles. Phys. Rev. Lett. 84, 4741–4744.Google Scholar
van Noije, T. P. C. & Ernst, M. H. 1998 Velocity distributions in homogeneous granular fluids: the free and the heated case. Granul. Matt. 1, 57–64.Google Scholar
Noskowicz, S. H., Bar-Lev, O., Serero, D. & Goldhirsch, I. 2007 Computer-aided kinetic theory and granular gases. Europhys. Lett. 79, 60001.Google Scholar
Ottino, J. M. & Khakhar, D. V. 2000 Mixing and segregation of granular materials. Annu. Rev. Fluid Mech. 32, 55–91.CrossRefGoogle Scholar
Pekeris, C. L. & Alterman, Z. 1957 Solution of the Boltzmann–Hilbert integral equation II. The coefficients of viscosity and heat conduction. Proc. Natl. Acad. Sci. USA 43, 998–1007.Google Scholar
Pouliquen, O., Delour, J. & Savage, S. B. 1997 Fingering in granular flows. Nature 386, 816–817.Google Scholar
Rao, K. K. & Nott, P. R. 2008 An Introduction to Granular Flow. Cambridge University Press.Google Scholar
Reinecke, S. & Kremer, G. M. 1990 Method of moments of Grad. Phys. Rev. A 42, 815–820.Google Scholar
Risso, D. & Cordero, P. 2002 Dynamics of rarefied granular gases. Phys. Rev. E 65, 021304.Google Scholar
Saha, S. & Alam, M. 2014 Non-Newtonian stress, collisional dissipation and heat flux in the shear flow of inelastic disks: a reduction via Grad’s moment method. J. Fluid Mech. 757, 251–296.Google Scholar
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41–74.Google Scholar
Sela, N., Goldhirsch, I. & Noskowicz, S. H. 1996 Kinetic theoretical study of a simply sheared two-dimensional granular gas to Burnett order. Phys. Fluids 8, 2337–2353.Google Scholar
Shukla, P. & Alam, M. 2009 Landau-type order parameter equation for shear banding in granular Couette flow. Phys. Rev. Lett. 103, 068001.Google Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos. Addison-Wesley.Google Scholar
Struchtrup, H. 2004 Stable transport equations for rarefied gases at high orders in the Knudsen number. Phys. Fluids 16, 3921–3934.CrossRefGoogle Scholar
Struchtrup, H. 2005 Macroscopic Transport Equations for Rarefied Gas Flows. Springer.CrossRefGoogle Scholar
Torrilhon, M. 2015 Convergence study of moment approximations for boundary value problems of the Boltzmann–BGK equation. Commun. Comput. Phys. 18, 529–557.Google Scholar
Torrilhon, M. 2016 Modeling nonequilibrium gas flow based on moment equations. Annu. Rev. Fluid Mech. 48, 429–458.Google Scholar
Umbanhowar, P. B., Melo, F. & Swinney, H. L. 1996 Localized excitations in a vertically vibrated granular layer. Nature 382, 793–796.Google Scholar
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